First, Azathoth, Yog-Sothoth and Nyarlathotep can be identified with the Holy Trinity. We know, from hints in HPL's work (most noticeably, in the Case of Charles Dexter Ward) and from Richard L. Tierney's "the Winds of Zarr" that Yog-Sothoth is YHWH, the Father. The Father is the transcendent aspect of the Trinity, the Creator, that precedes creation and existence as we understand it, beyond both time and space.

Then, Nyarlathotep is the Son: God manifested, God made present in the universe, the intervention of God in history and creation. He is the Word (fitting his role as the Messenger and agent of the Outer Gods), the instrument of creation and also the Firstborn, the first created. He is both Lucifer and Christ, an agent of change and transformation, bringing knowledge that can both save and condemn - and most likely both. He is Maia, who both weaves the illusion of existence that cloaks Brahma / Yog-Sothoth and works to reveal the truth and enlighten the worthy.

Azathoth would be the Holy Spirit, the immanent aspect of the Trinity. He exists at the center of creation. But an infinite universe has no center, or rather, everywhere is its center, so he exists everywhere. But this is also a metaphor: he exists at the center of time and space because he is the origin and cause of both. He is what manifests as both matter and energy, and his purest manifestation, the Court of Madness, is the Big Bang, where time and space, matter and energy find their origin and break down, meaningless, undifferentiated. Azathoth is the Primun Mobile and the Demiurge. His separation from Sophia / wisdom / meaning, Yog-Sothoth (who dwells at Keter, the fountain of all meaning, beyond the ultimate gate where the archetypes which are the model for all that exist dwell) created this reality, a pale reflection of the perfect ideals which are its model. This is why he is said to be an idiot: he is power without purpose or guide, without meaning o direction, absurdly imitating the pure forms of the Higher Realm. In the same way, humans are just pale reflections, faulty vessels (qelippot) of the souls they contain, reincarnating in different bodies in an endless quest for freedom and return to the source. This can only happen with the entry of yog-sothoth into this universe, which would be both its apotheosis and apocalypse, the reverse Tzimtzum, as the universe is reabsorbed into Keter whence it emanated, and then into the Ein Sof, where all differentiation disappears.